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The free energy of the two-dimensional dilute Bose gas. II. Upper bound

We prove an upper bound on the free energy of a two-dimensional homogeneous Bose gas in the thermodynamic limit. We show that for $a^2 ρ\ll 1$ and $βρ\gtrsim 1$ the free energy per unit volume differs from the one of the non-interacting system by at most $4 πρ^2 |\ln a^2 ρ|^{-1} (2 - [1 - β_{\mathrm{c}}/β]_+^2)$ to leading order, where $a$ is the scattering length of the two-body interaction potential, $ρ$ is the density, $β$ the inverse temperature and $β_{\mathrm{c}}$ is the inverse Berezinskii--Kosterlitz--Thouless critical temperature for superfluidity. In combination with the corresponding matching lower bound proved in \cite{DMS19} this shows equality in the asymptotic expansion.